The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: Fermat

Analytic geometry is the study of geometry using a coordinate system. Basically it’s the idea of expressing geometric objects such a as a line or a plane as an algebraic equation, think y=mx+b or ax+by+cz=k. This may be done by use of the more familiar Cartesian coordinates, by something such as polar coordinates or by just about any system for defining coordinates in a Euclidean space. The Common Core has the concept of graphing introduced in 5th grade, and graphing simple functions in the 8th grade. It’s quite interesting that something which took brilliant men so long to develop is now introduced to ten year olds.

The earliest evidence of anything resembling analytic geometry was by the Geek mathematician Menaechmus (380–320 BC), who was a student of Eudoxus and a tutor of Alexander the Great. Proclus and Eutocius both report that Menaechmus discovered the ellipse, hyperbola and parabola and that these were initially called the “Menaechmian triad”. These were used along with something resembling analytic geometry to solve the Delian problem, which is to, given the edge of a cube to construct the edge of a cube with double the volume. Though most of what we know of Menaechmus and his exact solution is second hand as his original work was lost, it appears as though he argued his solution for doubling the cube with proportions of a side length to the area of a side which fairly quickly leads to conics.

Another early manifestation of analytic geometry was by Omar Khayyám, whom we have mentioned in class. He drew a connection between algebra and geometry in his solution of general cubic equations. His idea to do this was to create a geometrical construction of a cubic equation by considering the variable to be the edge of a cube and constructing a set of curves from which a solution could be discerned. While it might seem far flung from Cartesian coordinates it was a significant leap in connecting the separate concepts of algebra and geometry.

Analytic geometry was more or less formalized in the early 17th century independently by René Descartes and Pierre de Fermat. Descartes published first and so he is commonly credited as the sole creator which leads to analytic geometry often being call Cartesian geometry. As Fermat has already been much discussed, I’ll skip his background and instead jump to Descartes. René Descartes was a French mathematician and philosopher who is most well known as the (co-)creator of analytic geometry and as the father of modern philosophy. He is the origin of the well-known quote “Je pense, donc je suis” or “I think, therefore I am” which appeared in in Discours de la methode (Discourse on the Method).

While the Fermat and Descartes constructions are equivalent, they did differ in several ways which primarily stem from which direction their creator worked. Fermat started with the algebraic equation and described the analogous geometric curve while Descartes worked in reverse, starting with the curve and finding the equation. To contrast the methods, the way most of us learn analytic geometry is much more similar to Fermat than to Descartes, where we learn to recognize that a degree 1 polynomial will represent a straight line then we learn how to find that line, next that quadratic function represents a parabola and so on. Whereas if we were to learn as Descartes’ work, we would take a straight line then learn that it represented a degree 1 polynomial which is similar to Fermat. But then working further in this direction, it doesn’t make sense to jump to parabolas and instead to talk about conics and all degree 2 polynomials with no reason to talk specifically about parabolas.

In 1637, Descartes published his method of connecting arithmetic, algebra, and geometry in the appendix La géométrie (The Geometry) of Discourse on the Method. However, given Descartes’s opaque writing style (to discourage “dabblers”) as well as The Geometry being written in French rather than in the more common (for academic purposes) Latin, the book was not very well received until it was translated into Latin in 1649, by Frans van Schooten, with the addition of commentary clarifying certain arguments. Interestingly, though Descartes is credited with the invention of the coordinate plane, since he describes all necessary concepts, no equations are in fact graphed in The Geometry and his examples used only one axis. It was not until its translation into Latin that the concept of 2 axes was introduced in Schooten’s commentary.

One of the most important early uses for analytic geometry was to help prove the validity of the heliocentric theory of planetary motion, the (then) theory that the planets orbited around the Sun. As analytic geometry was one of the first methods one could use to actually make computations about curves, it was used to model elliptical orbits so as to demonstrate the correctness of this theory. Analytical geometry, and particularly Cartesian coordinates, were instrumental in the creation of calculus. Just consider how you might calculate something like the “area under the curve” without the concept of the curve being described by some algebraic equation. Similarly, the idea of rate of change of as function of time at a particular time becomes much clearer when thought of as the slope of the tangent line, but to do this, we need to think of the function as having some representation in the plane for which we need analytic geometry.

Every mathematician, or student of the subject, has heard of Pierre de Fermat’s last theorem, but surely such a talented mathematician has more to show for himself than a proof that he didn’t share with anyone. Fermat’s life and personality is interesting in and of itself, so while this written analysis is primarily a recognition of his other accomplishments, it can only be made more interesting by mixing in background on who Fermat was. While his other contributions to mathematics may not satisfy humanity’s craving for drama, Fermat did contribute more to mathematics than his most famous challenge.

Fermat was heavily influenced by François Viète, a French mathematician and lawyer. Fermat’s methodology in mathematics is described as classical Greek with apparent influence from Viète, and because his methodology consists of sending theorems to his buddies via letter with little to no proof attached, one could only assume the Greeks are responsible for his desire for arrogant competition, and the tender romanticism of the now lost art of letter-writing came from France. Although Fermat claimed that the proof for all his theorems was in the pudding, no one can find the pudding. Other credible mathematicians such as Karl Gauss had their doubts, but there is something particularly entertaining about a man flaunting claims in his peer’s faces and challenging them to prove something that he may or may not have. It’s a little suspicious, did he prove his theorems and simply enjoy rubbing his friend’s inability to in their faces? Well he was a lawyer so it’s possible. Did he trick (or inspire depending on how positively you like to think) other mathematicians into proving things that he couldn’t?

Fermat Invented analytic Geometry and contributed to the development of Calculus inspiring other magnificent minds such as Isaac Newton. In fact Newton admitted that some of his early ideas came from “Fermat’s way of drawing tangents” (Pierre De Fermat, Wikipedia). A manuscript of Fermat’s was published in about 1679 in Varia Opera Mathematica (Pierre De Fermat, Wikipedia. The title of his is manuscript “Ad Locos Planos et Solidos Isagoge” which is Latin, one of many languages that Fermat was conversant in including Greek, Italian and Spanish. If you translate that to English it reads an introduction to plane solid loci. The text can be more specifically interpreted as a classification of curves as: plane, solid or linear. Plane curves being straight lines and circles, solid curves being ellipses, parabolas and hyperbolas and linear curves being described kinematically with some sort of condition (Ad Locos Planos Et Solidos Isagoge).

Moving on to differential calculus, Fermat developed a technique known as adequality that he used for determining maxima, minima and tangents to a curve. Adequality can be defined as approximately equal and is denoted with the ‘~’ symbol (Adequality, Wikipedia). Basically Fermat would compare a function, say f(x), to something approximately the same, say f(x+Ɛ). He’d set them equal to each other which is of course risky and Fermat was likely under oath, so it must be the case that he was being meticulously careful not to be held in contempt of the court when he decided to instead set them adequal. Then by canceling out like terms, dividing by Ɛ and “solving for x” he’d derive his value for a maxima or minima.

The Fundamental Theorem in Calculus is arguably one of the most important concepts in mathematics and Fermat had a hand in inspiring it. By evaluating the integral of general power functions Fermat produced a formula that ended up being useful to Newton when he developed the fundamental theorem of calculus. Fermat evaluated such integrals by reducing them to a sum of geometric series.

In 1654 Fermat teams up with Blaise Pascal. Fermat and Pascal collaborated on a classical probability problem known as the problem of points. The problem is a game of sorts. It has two players. Each player has equal chance to win each round. There is a prize pot and the players agree that a particular number of round wins leads to total victory. The game is interrupted and of course that leaves the question of who gets what amount of the prize. The concept of a “fair” division must be established based on how many rounds each player has won so far and the probability that that player was going to win the overall pot. Fermat figured out how many possibilities were left based on how many rounds were left and charted these possibilities then based his “fair” division in proportion to the probability of these possibilities. Fermat’s solution was inefficient and inaccurate as the number of rounds left gets large and so Pascal made some big improvements on it. Nonetheless this collaboration earned Fermat and Pascal the title of founders of probability theory on the grounds that this game laid the groundwork for probability theory (Pierre De Fermat, Wikipedia).

Yes Fermat’s Last Theorem is his most notable and interesting accomplishment, but Fermat contributed much more to the world of mathematics than this. He inspired other great minds with his own work, irritated others with his arrogance and sass, and helped laid the ground work for some of the biggest mathematical concepts by getting his hands dirty with proofs. Even though his pudding will likely never be found, his proofs/challenges (or lack thereof) inspired or tricked other mathematicians into some of the greatest discoveries in mathematics and some less magnificent discoveries as well.

It is not often a person contributes to a field they do not even work in the way Pierre de Fermat has contributed to the field of mathematics. Born to a wealthy leather merchant, Fermat received a bachelor’s in civil law from the University of Orléans and went on to become a lawyer, while at the same time engraving his name into math history books, doing said math just for recreation. His importance in mathematics lead to many theorems named after him, as well as numbers. These numbers are known as Fermat numbers, which are positive integers, that take of the form Fn = 2(2n) + 1, when n is nonnegative and an integer. For example for F1, F1= 2(2)+1= 5. The first five Fermat numbers are 3, 5, 17, 257, and 65537, and these numbers continue to grow to incredibly large magnitudes. Fermat believed this form created an infinite number of prime numbers, which are known as Fermat primes.

Fermat numbers are occasionally written as 2n+1, but since when n is greater than zero and Fn prime, n must be a power of two, the form Fn = 2(2n) + 1 is the common form for Fermat numbers. One of the main problems with Fermat claiming all these numbers are prime is the fact that they soon become too large to calculate for even today’s computers, let alone a man with his pen and paper in the 17th century. Unfortunately for Fermat, by the time the 18th century rolled around, he was dead. In 1732, mathematician Leonhard Euler found that F5, which is 4,294,967,297, is actually divisible by 641, most likely figuring this out from having a large amount of time on his hands. While this showed that some Fermat numbers are not actually prime, excluding when n=0 in the form 2n+1, it does not discount the fact that the Fermat number equation could still make an infinite number of primes, since there are infinite amount of Fermat numbers. However as of now, the only Fermat primes that are known are F0 through F4.

Now initially I found the idea of an equation, the equation here being Fn = 2(2n) + 1, that finds only certain prime numbers, most of which are way too large to even be calculated even 400 years after the equation for them was created, the equivalent to a student doing extra credit when he has a 98% in his class. What I’m trying to say is, I found Fermat numbers pointless and to be the 17th century mathematician’s version of a braggadocio. However, I know nothing and Gauss managed to find a relationship between “Euclidean construction of regular polygons and Fermat primes,” where he showed a regular 17-gon could be constructed. It was also found a regular n-gon can be created if n is the product of any number of Fermat primes and the number 2. These regular n-gons take the property of being able to be constructed with a compass and straightedge. Who would have thought one of the greatest mathematicians to ever live could leave me feeling so inadequate, at least mathematically.

Similar to Fermat numbers are what are known Mersenne numbers, created by 17th century French mathematician and music theorist Marin Mersenne. Yet again a person being a jack of all trades, except instead of being a master of none they were a master of a few or at least of one. Mersenne numbers take of the form Mn = 2n -1, and Mersenne primes are numbers that take that form which are prime. Mersenne believed that for n<=257, Mn was prime for n= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and the rest are composite. While this belief turned out to incorrect, he still got the name for the primes. Just like Fermat primes, it is unknown whether there are an infinite number of Mersenne primes, but as of now 48 Mersenne primes are known, the largest being 257885161-1, which again makes me wonder how much time do some of these mathematicians have on their hands.

Mersenne numbers were originally studied because of their connection to perfect numbers, which are positive integers that are equal to the sum of their divisors. Euclid proved that if the number 2n-1 is prime, then 2n-1(2n-1) is a perfect number, which many years later led to Euler discovering that all even perfect numbers come in this form. Another interesting fact is that the ten largest known prime numbers are Mersenne numbers. I personally find number theory incredibly interesting, partly because I like numbers and partly because how mathematicians are able to come with these theorems and proofs baffle me. I ultimately wonder if they had any true goals when thinking about these primes, or if it was just for the pure fun and interest in it.

After reading several different sources of math history, I was attracted by Fermat’s history and his theorems about prime numbers. I realized that I am kind of interested in these numbers and anything related to these numbers after digging into the computer science major for few years. Why? I think the primary reason is because prime numbers are widely used in computer science for security. For example, the RSA algorithm, which is a public-key crypto system, uses prime numbers to generate public keys. If you are not familiar with the RSA algorithm, I will explain a little bit more here. Basically, the RSA algorithm generates a public key based on two large prime numbers. The prime numbers are secret. Anybody is able to use the public key to encrypt a message. When I first learned this RSA algorithm in my algorithm class, I felt like it was magic! After taking computer networking, the computer networking class, I realized that prime numbers were everywhere in my life. For example, you go to amazon.com to buy some stuff, you need to login to your account and pay the bill online. Behind the scenes, there are prime numbers securing your account and transactions. This interest brought a new question to me, which was how to generate a big prime number? Another way to think about this question is how to determine whether a number is a prime?

FLT and modern computer science. Image by the author.

Because of my curiosity about prime numbers, I enjoyed reading and thinking about FLT. In class, FLT means Fermat’s Last Theorem, but in this blog post, it means Fermat’s Little Theorem. Basically, FLT says if p is a prime number, then for any integer a, a^p – p is a multiple of p. If a is not divisible by p, then a^(p-1) – 1 is an integer which is multiple of p (Wikipedia). How can this math theorem help a computer science student? I solved some programming questions related to prime numbers. For example, generate all prime numbers less than n. If n is 10, my program should return 2, 3, 5, 7 as a result. The other programming question was to write a program that runs a primality test, which is used to see whether a number is prime. For example, if the input number is 10, my program should return true if 10 is a prime number and false if 10 is not a prime number. I was able to solve such questions by using an inefficient algorithm. If a number was very big, it took more than an hour to get the final result. Obviously, I did not expect such a slow algorithm. By combining FLT into programming, I can write an efficiency algorithm to solve those questions.

Computer science majors might be wondering how a Fermat primality test works. First of all, we have an integer n and we need to choose some integers co-prime to n. Then we need to calculate a^(n-1) mod n. Let’s say the result is different from 1, then n is composite. If the result is 1, then we can say n may or may not be a prime number. For example, let’s use p = 341 and a = 2, then we have 2^340 = 1 (mod 341). Obviously, 341 is not a prime number because 341 = 11 * 31. So we call 341 a pseudoprime base 2. Such a pseudoprime number is also called a Carmichael number. This is a primary reason why we cannot directly use the FLT for a primality test: it will be fooled by some Carmichael numbers. That means we have to do something else to get it work on solving programming questions! There are only 21853 pseudoprimes base 2 for n from 1 to 25 * 10^9. How does this data help us? If 2^n – 1 mod n = 1, then n is a prime number, or n is one of those 21853 pseudoprimes. I think we can build a list of pseudoprimes before running our program for primality test. When we need to check if a number is a prime number, we can check if this number is a pseudoprimes in the list. If it is not one of the pseudoprimes, we can start running the program for primality test. The FLT was like the roots of a binary tree. People used this root to branch out its left children and right children. In our case, one of the children referred to is the Fermat primality test. At this moment, I cannot stop digging deeper in this tree. I want to traverse the whole tree just like running a breadth-first search in my brain.

Number theory is a mystery! After reading the articles, I asked myself a question “How many unknown theorems still exist?” It might be like the universe – people have only explored a tiny part of it. It may hide more secrets behind the scenes – people need time to reveal those secrets. Probably after a few years, scientists will discover many more theorems, just like FLT.

This xkcd comic has two points. The first is understandable without any context. If the writer had in fact discovered a proof that information is infinitely compressible, then ANY amount of space would be sufficient to contain it. The second point refers to Fermat’s famous statement “I have discovered a truly remarkable proof of this theorem which this margin is too small to contain,” which was, of course, referring to Fermat’s Last Theorem, a topic which we discussed extensively in class.

Liberal use of others property

It is now often believed that Fermat did not actually have a correct proof of this theorem. This minor detail did not, however, deter the great Fermat from writing it as fact in the margin of his copy Arithmetica to be discovered posthumously and baffle mathematicians for centuries to come. This, however, is not the only case of mathematicians writing statements in strange places. Another mathematician who did this was William Rowan Hamilton. Unlike Fermat, Hamilton decided to actually carve in his answer to a question, as opposed to carving in a claim that he has an answer. To be fair though, Fermat did own his book, while Hamilton didn’t actually own the bridge. This occurred in 1843, while taking a walk, he had a flash of brilliance during which he discovered Quaternions. Lacking a proper way with which to write down the result, Hamilton instead chose to carve his answer in the side of a bridge.

But what are quaternions?

Hamilton knew how to add and multiply complex numbers in a plane. However, he did not know how to multiply them in space. Quaternions were his solution to this problem, because while he could not figure out how to multiply complex points in a 3-dimensional space, he could figure out how to do it in a 4-dimensional space. In fact there is now a theorem which says the only normed division algebras which are number systems where we can add, subtract, multiply, and divide, and which have a norm satisfying |zw|=|z||w| have dimension 1, 2, 4, or 8. Quaternions can be thought of as a 4-dimensional space and are often denoted by H or ℍ. They are a noncommutative number system over the complex space, which just means that a*b does not necessarily equal b*a. They are defined as ℍ ={a+bi+cj+dk} where a, b, c, and d all belong to the real numbers. Note in particular that ij = k = -ji, jk = i = -kj, and ki = j = -jk. This eventually leads to what Hamilton engraved on the Brougham Bridge: i2 = j2 = k2 = ijk = -1, which means that i, j, and k are all equal to square root of -1.

Utility of quaternions

The quaternions can be used to do rotations in 3 dimensions, which may seem unintuitive given that quaternions describe a 4-dimensional space. To better explain this we need the concept of real and pure quaternions. A real quaternion is one which contains only a real part, while a pure quaternion is one which does not contain a real part. This is the equivalent of partioning a complex number into its real and imaginary parts. The difference between these two scenarios is that the pure portion of a quaternion is a vector in 3-space instead of a single number. Thus a real quaternion will take the form [a, 0] where 0 is the zero vector and a pure quaternion will take the form [0, v] where v is a vector of the form v=bi+cj+dj. Note that this means that the set of all pure quaternions define a 3-space. Thus the process of rotating in three dimensions is accomplished by starting with a pure quaternion, called p. This quaternion is then multiplied by the rotor, a second quaternion, called q, of the form [cos(Θ), sin(Θ)*v] where v is a vector of the form v=bi+cj+dj and Θ is the angle by which we are rotating. If p happens to be perpendicular to q then the result will be a pure quaternion and the process is complete. However, if it is not the resulting quaternion will not be pure and the magnitude will be off. We can, however, multiply this new result by the inverse of q which will result in a pure quaternion of the desired length. Note that this means that the object should start and end in the 3-dimensional space as defined by the set of all pure quaternions with the real portion being used as an intermediary. I should also mention that the inverse is the conjugate of the quaternion divide by its normalization squared, where the conjugate is computed by negating the vector v and the normalization by dividing by the magnitude of the quaternion. Quaterions, however, don’t just allow for rotation in 3 dimensions, they also help avoid certain problems such as gimbal lock. Gimbal lock occurs when two out of the three rotational axes align. When this happens, the aligned axes both rotate the object in the same way. While you can still get out of the gimbal lock, it does force you to do some additional rotations. Quaternions circumvent this problem by having that intermediary 4th rotational axis.

Conclusion

If you want to commit vandalism, all you have to do is discover something brilliant which will be used for quite some time after its discovery in technologies which have yet to exist and engrave it in the side of a bridge or scribble it within the margins of a book. You might even get a plaque commemorating your vandalism.

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form. Fn=22n+1, where n is a nonnegative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617. And he is the first to investigate numbers of the form 22n.

There is the Pépin test which gives sufficient and necessary condition for the primality of the Fermat prime and this can only be implemented by use of modern computers. Pépin’s test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth’s test. The test is named for a French mathematician, Théophile Pépin.Let Fn be a Fermat number. Fn is prime if and only if 3(Fn-1)/2 = -1 (mod Fn).Here 3 can be replaced by any positive integer k for which the Jacobi symbol (k|Fn) is -1. These include k=3, 5, and 10.If Fn is prime, this primality can be shown by Pepin’s test, but when Fn is composite, Pepin’s test does not tell us what the factors will be (only that it is composite). For example, Selfridge and Hurwitz showed that F14 was composite in 1963, but we still do not know any of its divisors. (Chris K. Caldwell)

Mersenne numbers, which take the form of 2n-1, were named after Marin Mersenne, a French monk from the early 17 century, who corresponded with Fermat. We are particularly interested in the case when Mersenne number are prime. It is not doubt that the first 17 primes of the form 2n-1 match the following n values: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281 (Zegarelli 287).

The first 12 Mersenne primes were known since 1914 and the 12th, 2127-1, was established by Anatole Lucas in 1876. It was one of the largest-known prime number for over 75 years. The next 5 Mersenne primes (p=13 to 17)were discovered in 1952. It was in 1952 when the testing program for Mersenne numbers was began. This led to the establishment of three other primes. More testing has been carried out using modern day computers and the smallest Mersenne number that is untested is 22309-1 (~2013), and this has not been a case of great interest. There is a conjecture that 2n-1 is always prime when n is a Mersenne number. And the more interesting case is the 28191-1 because the 8191 also is the Mersenne number. (Křížek, Florian and Lawrence 214).